Question

The differential operator (D+ 8)³ is supposed to annihilate the function x²e⁻⁸ˣ, we will check that: (D)(x²e⁻⁸ˣ) = 8xe⁽⁻⁸ˣ⁾(1-x) (D+8)(x²e⁻⁸ˣ) = 8xe⁽⁻⁸ˣ⁾ D((D+8)(x²e⁻⁸ˣ)) = -64xe⁽⁻⁸ˣ⁾+8e⁽⁻⁸ˣ⁾ (D+8)²(x²e⁻⁸ˣ) = 8e⁽⁻⁸ˣ⁾ D((D+8)2(xe⁻⁸ˣ)) = -64e⁽⁻⁸ˣ⁾ (D+8)³(x²e⁻⁸ˣ) = 0

Answer 1

To check if the differential operator (D+ 8)³ annihilates the function x²e⁻⁸ˣ, we substitute the function into the operator and simplify the expression step by step.

Step-by-step explanation:

To check if the differential operator (D+ 8)³ annihilates the function x²e⁻⁸ˣ, we can substitute the function into the operator. Using the operator (D) which represents the derivative, we find (D)(x²e⁻⁸ˣ) = 8xe⁽⁻⁸ˣ⁾(1-x). Next, substituting the result into (D+8) gives (D+8)(x²e⁻⁸ˣ) = 8xe⁽⁻⁸ˣ⁾. Now, we can take the derivative of this expression using (D) again, resulting in D((D+8)(x²e⁻⁸ˣ)) = -64xe⁽⁻⁸ˣ⁾+8e⁽⁻⁸ˣ⁾. Finally, substituting the expression into (D+8)² and (D+8)³ will yield the respective results.